Question

Show transcribed image textYou are given the initial value problem y" + y = 0, y(0) = 1, y'(0) = -1, and are assured that there is a solution, and that it has the form y(x) - ci middot cos(x) 4- C2 middot sin(i). Decide what the constants c1 and c2 must be, and verify that the resulting function actually is a solution to the initial value problem for all values of x. Now solve the general problem y" + y = 0, y(0) = a, y'(0) = b, where a and b arc some real numbers. A reasonable guess, based on part (a), is to look at functions of the form c1 middot cos(x) + C2 middot sin(x); try to express c1 and C2 in terms of a and b. Can you show that the initial value problem y" + y = 0, y(0) = a, y'(0) = b, has exactly one solution? Is it possible that there are other solutions that you haven't thought of vet? Ex­plain as thoroughly as you can.